LESSON
Page 1 of 5
9.4
Real Numbers
BEFORE
You ordered rational
numbers.
Vocabulary
irrational number,
p. 470
real number, p. 470
Now
WHY?
You’ll compare and order real
numbers.
So you can compare the periods
of two pendulums, as in Ex. 34.
In Lesson 5.1, you learned that a rational number is a number that can
be written as a quotient of two integers. All rational numbers have
decimal forms that terminate or repeat.
An irrational number is a number that cannot be written as a quotient
of two integers. The decimal form of an irrational number neither
terminates nor repeats. For example, in the irrational number below, the
pattern of ones separated by an increasing number of zeros continues
without end. The decimal neither terminates nor repeats.
0.1010010001000010000010000001. . .
The real numbers consist of all rational and irrational numbers. The
Venn diagram shows the relationships among the real numbers.
Real Numbers
Rational numbers
Integers
Irrational numbers
Whole numbers
Example 1
Number
Decimal Form
Decimal Type
Type
5
5
0.625
8
Terminating
Rational
b. 6
5
5
0.83333. . . 0.83
6
Repeating
Rational
c. 19
19
≈ 4.35889894. . .
Study Strategy
Notice in part (c) of Example 1
is irrational. The
that 19
square root of any whole
number that is not a perfect
square is irrational.
Classifying Real Numbers
a. 8
Nonterminating, Irrational
nonrepeating
Checkpoint
Tell whether the number is rational or irrational.
2
1. 3
2. 100
3. 6
4.
16
25
5. Critical Thinking Consider the positive square roots of the whole
numbers from 1 to 10. What percent of these numbers are irrational?
470
Chapter 9
Real Numbers and Right Triangles
Page 2 of 5
Review Help
For help with using a compass,
see p. 792.
Graphing Irrational Numbers You can graph an irrational number on a
, draw a right triangle with one leg
number line. For instance, to graph 2
on the number line and each leg with a length of 1 unit, as shown. By the
Pythagorean theorem, the length of the hypotenuse is 12 12 2
.
2
Use a compass to
transfer the length
of the hypotenuse
to the number line.
1
0
1.0
2
2.0
Notice that the graph of 2
is close to the graph of 1.4. So, 2
≈ 1.4,
, which is 1.41421356. . . .
which agrees with the decimal form of 2
Example 2
Comparing Real Numbers
9
5
Copy and complete 2
_?_ using <, >, or .
95
2
0
1
2
is to the left of 9.
5
2
9
5
Answer 2
< Example 3
Study Strategy
Another Way You can
compare and order real
numbers by using a calculator
to find decimal forms of the
numbers, then comparing and
ordering the decimals.
Ordering Real Numbers
4
Use a number line to order the numbers , 2.8, 3
, and 5
from
3
least to greatest.
Graph the numbers on a number line and read them from left to right.
2.8
43
5
3
2
1
0
1
3
2
4
3
, , and 3
.
Answer From least to greatest, the numbers are 2.8, 5
Checkpoint
Copy and complete the statement using <, >, or .
8
6. 7
_?_ 3
3
7. _?_ 2
2
8. 32
_?_ 4.2
15
9. _?_ 15
4
Use a number line to order the numbers from least to greatest.
7
10. 2.9, 10
, , 22
3
16
11. , 22
, 32
, 4.6
3
Lesson 9.4
Real Numbers
471
Page 3 of 5
Example 4
Using Irrational Numbers
Landmark Buildings Your class is visiting historical landmarks in
Chicago. Outside the Washington Block, you break up into two
groups. Group A walks about 800 meters east and 200 meters south
to the Chicago Building. Group B walks about 600 meters south and
200 meters east to the Rookery Building. To the nearest 10 meters, how
much farther is group A from the Washington Block than group B is?
Solution
Draw a diagram. Then use the
Pythagorean theorem to find each
distance from the Washington Block.
8002 2002 680,00
0
Group A: ≈ 825
N
Washington Group A
800 m
Block
200 m
?
600 m
2
Group B: 600
2002 400,00
0
≈ 632
Chicago
Building
?
Group B
Rookery Building
200 m
Difference in distances: 825 632 193
Answer To the nearest 10 meters, group A is about 190 meters farther
from the Washington Block than group B is.
9.4
Exercises
INTERNET
More Practice, p. 811
CLASSZONE.COM
eWorkbook Plus
Guided Practice
Vocabulary Check
1. Explain what an irrational number is. Give an example.
2. Where would the number 7.5
2
appear in the Venn diagram on
page 470? Explain your thinking.
Skill Check
Tell whether the number is rational or irrational.
2
3. 7
4. 49
5. 71
6. 8.3
4
Copy and complete the statement using <, >, or .
7. 7
_?_ 2
8. 11
_?_ 9
8
9. 16
_?_ 3
11. Fences A wire fence with wooden posts and
rails has the dimensions shown. Each section
of fence has a diagonal support brace as
shown. Is the exact length of the brace a
rational or an irrational number of feet?
472
Chapter 9
Real Numbers and Right Triangles
5
10. _?_ 2
3
8 ft
4 ft
Page 4 of 5
Practice and Problem Solving
Homework Help
Example
1
2
3
4
Exercises
12–19, 28–32
20–23
24–27
33–35
Online Resources
CLASSZONE.COM
• More Examples
• eTutorial Plus
Tell whether the number is rational or irrational.
3
12. 4
16.
16
5
13. 8
14. 81
16
15. 5
17. 14.4
18. 17.6
5
19. 10.1
Copy and complete the statement using <, >, or .
5
20. _?_ 6.25
2
27
21. 43
_?_ 4
22. 32
_?_ 5.6
23. 0.5
_?_ 0.5
Use a number line to order the numbers from least to greatest.
19 1
24. 3.5, 23
, 13
, , 3 , 8
5
4
9
25. 4
, 5
, 0, , 2, 3
5
17
3
26. , 8.6, 64
, 36
, 7 , 50
2
4
67
25
27. 4, 18
, , 4
6
Copy and complete the statement using always, sometimes, or never.
Explain your reasoning.
28. A negative integer is _?_ a whole number.
29. A square root of a positive number is _?_ an irrational number.
30. A real number is _?_ a rational number.
31. A whole number is _?_ an irrational number.
32.
Writing
The area of a square is 7 square meters. Is the perimeter of
the square a rational or an irrational number of meters? Explain.
33. Geometry A rectangle is twice as long as it is wide, and it has an area of
20 square meters.
a. Let w represent the width of the rectangle. Write a variable
expression in terms of w for the length of the rectangle.
b. Use the formula for the area of a rectangle to write an equation for
the area of the given rectangle.
c. Find the width of the rectangle to the nearest tenth of a meter.
d. Find the length of the rectangle to the nearest tenth of a meter.
34. Pendulums The period of a pendulum is the time that
it takes the pendulum to swing from one side to the
other and back. A pendulum’s period P (in seconds)
and its length l (in feet) are related by the equation
P 1.1l .
Pendulum in the Museum of
Science in Boston,
Massachusetts
l
The giant pendulum in the Science Museum of
Virginia in Richmond is about 96 feet long. The
giant pendulum in the New Detroit Science Center
in Detroit is 40 feet long. How much longer, to the
nearest second, is the period of the pendulum in
Richmond than the period of the pendulum in Detroit?
Lesson 9.4
Real Numbers
473
Page 5 of 5
35. Sailing The maximum speed at which a boat can travel is called its hull
speed. The hull speed h (in nautical miles per hour) of some boats is
, where l is the length in feet of the boat
given by the equation h 1.8l
at the water line. One boat is 40 feet long at the water line. A second
boat is 60 feet long at the water line. Which boat has a faster hull
speed? How much faster? Give your answer to the nearest nautical mile
per hour.
Name an irrational number between the given rational numbers.
36. 1 and 2
37. 4 and 3
1
38. 15 and 15 2
39. 10.1 and 10
40. Error Analysis Your friend says that 2
is rational because 2
can be
22
written as the fraction . Explain your friend’s error.
2
41. Critical Thinking Is the product of two irrational numbers always,
sometimes, or never irrational? Give examples to support your answer.
42. Extended Problem Solving Use the following method to draw the
Wheel of Theodorus. The first three triangles in the wheel are shown.
1
a. Draw a Diagram Start near the center of a large
sheet of paper. Draw a right triangle with legs
1
1 unit long. Draw the next right triangle using the
? ?
?
hypotenuse of the previous triangle as one leg, and
1
a length of 1 unit for the second leg. Repeat this
process to draw at least six triangles.
1
This quilt is entitled Wheel of
Theodorus.
b. Calculate Beginning with the first triangle you drew, find the length
of the hypotenuse of each triangle in simplest form. Label the length
of each hypotenuse on your drawing.
c. Critical Thinking Make a list of the hypotenuse lengths in your
drawing. Describe the numbers in the list.
43. Challenge Show how to graph 34
on a number line using a right
triangle. Explain your method.
Mixed Review
Evaluate the expression when a 1.5. (Lesson 1.2)
44. a 2
45. a 3
46. a 4
Find the slope of the line through the given points. (Lesson 8.4)
47. (5, 2), (4, 3)
48. ( 3, 7), (1, 5)
49. (6, 2), (4, 8)
Evaluate the expression when a 26 and b 10. (Lesson 9.1)
50. a
b
Standardized Test
Practice
51. a
b
52.
b 2 (a
10)
53. Extended Response You are building four corner shelves, each in the
shape of a right triangle. Each leg of each triangle is 14 inches long. You
plan to cover the longest edge of each shelf with a strip of decorative
trim that can be purchased only by the foot.
a. Explain how to estimate the number of feet of trim you need to
purchase for all four shelves.
b. Find the exact amount of trim you need to use for all the shelves.
474
Chapter 9
Real Numbers and Right Triangles